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Whilst a modest number of investigations have been undertaken concerning nanofluids (NFs), the exploration of fluid flow under exponentially stretching velocities using NFs remains comparatively uncharted territory. This work presents a distinctive contribution through the comprehensive examination of heat and mass transfer phenomena in the NF ND–Cu/H₂O under the influence of an exponentially stretching velocity. Moreover, the investigation delves into the intriguing interplay of gyrotactic microorganisms and convective boundary conditions within the system.

Similarity transformations have been used on PDEs to convert them into dimensionless ODEs. The solution is derived by using the homotopy analysis method (HAM). The pictorial notations have been prepared for sundry flow parameters. Furthermore, some engineering quantities are calculated in terms of the density of motile microbes, Nusselt and Sherwood numbers and skin friction, which are presented in tabular form.

The mixed convection effect associated with the combined effect of the buoyancy ratio, bioconvection Rayleigh constant and the resistivity due to the magnetization property gives rise to attenuating the velocity distribution significantly in the case of hybrid nanoliquid. The parameters involved in the profile of motile microorganisms attenuate the profile significantly.

The current simulations have uncovered fascinating discoveries about how metallic NFs behave near a stretched surface. These insights give us valuable information about the characteristics of the boundary layer close to the surface under exponential stretching.

The novelty of the current investigation is the analysis of NF ND–Cu/H₂O along with an exponentially stretching velocity in a system with gyrotactic microorganisms. The investigation of fluid flow at an exponentially stretching velocity using NFs is still relatively unexplored.

The need for precise synthesis of customized designs has resulted in the development of advanced coating processes for modern nanomaterials. Achieving accuracy in these processes requires a deep understanding of thermophysical behavior, rheology and complex chemical reactions. The manufacturing flow processes for these coatings are intricate and involve heat and mass transfer phenomena. Magnetic nanoparticles are being used to create intelligent coatings that can be externally manipulated, making them highly desirable. In this study, a Keller box calculation is used to investigate the flow of a coating nanofluid containing a viscoelastic polymer over a circular cylinder.

The rheology of the coating polymer nanofluid is described using the viscoelastic model, while the effects of nanoscale are accounted for by using Buongiorno’s two-component model. The nonlinear PDEs are transformed into dimensionless PDEs via a nonsimilar transformation. The dimensionless PDEs are then solved using the Keller box method.

The transport phenomena are analyzed through a comprehensive parametric study that investigates the effects of various emerging parameters, including thermal radiation, Biot number, Eckert number, Brownian motion, magnetic field and thermophoresis. The results of the numerical analysis, such as the physical variables and flow field, are presented graphically. The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as fluid parameter increases. An increase in mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid.

Intelligent materials rely heavily on the critical characteristic of viscoelasticity, which displays both viscous and elastic effects. Viscoelastic models provide a comprehensive framework for capturing a range of polymeric characteristics, such as stress relaxation, retardation, stretching and molecular reorientation. Consequently, they are a valuable tool in smart coating technologies, as well as in various applications like supercapacitor electrodes, solar collector receivers and power generation. This study has practical applications in the field of coating engineering components that use smart magnetic nanofluids. The results of this research can be used to analyze the dimensions of velocity profiles, heat and mass transfer, which are important factors in coating engineering. The study is a valuable contribution to the literature because it takes into account Joule heating, nonlinear convection and viscous dissipation effects, which have a significant impact on the thermofluid transport characteristics of the coating.

The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as the fluid parameter increases. An increase in the mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid. Increasing the strength of the magnetic field promotes an increase in the density of the streamlines. An increase in the mixed convection parameter results in a decrease in the isotherms and isoconcentration.

The purpose of this paper is to study haemodynamic flow characteristics and entropy analysis in a bifurcated artery system subjected to stenosis, magnetohydrodynamic (MHD) flow and aneurysm conditions. The findings of this study offer significant insights into the intricate interplay encompassing electro-osmosis, MHD flow, microorganisms, Joule heating and the ternary hybrid nanofluid.

The governing equations are first non-dimensionalised, and subsequently, a coordinate transformation is used to regularise the irregular boundaries. The discretisation of the governing equations is accomplished by using the Crank–Nicolson scheme. Furthermore, the tri-diagonal matrix algorithm is applied to solve the resulting matrix arising from the discretisation.

The investigation reveals that the velocity profile experiences enhancement with an increase in the Debye–Hückel parameter, whereas the magnetic field parameter exhibits the opposite effect, reducing the velocity profile. A comparative study demonstrates the velocity distribution in Au-CuO hybrid nanofluid and Au-CuO-GO ternary hybrid nanofluid. The results indicate a notable enhancement in velocity for the ternary hybrid nanofluid compared to the hybrid nanofluids. Moreover, an increase in the Brinkmann number results in an augmentation in entropy generation.

This study investigates the flow characteristics and entropy analysis in a bifurcated artery system subjected to stenosis, MHD flow and aneurysm conditions. The governing equations are non-dimensionalised, and a coordinate transformation is applied to regularise the irregular boundaries. The Crank–Nicolson scheme is used to model blood flow in the presence of a ternary hybrid nanofluid (Au-CuO-GO/blood) within the arterial domain. The findings shed light on the complex interactions involving stenosis, MHD flow, aneurysms, Joule heating and the ternary hybrid nanofluid. The results indicate a decrease in the wall shear stress (WSS) profile with increasing stenosis size. The MHD effects are observed to influence the velocity distribution, as the velocity profile exhibits a declining nature with an increase in the Hartmann number. In addition, entropy generation increases with an enhancement in the Brinkmann number. This research contributes to understanding fluid dynamics and heat transfer mechanisms in bifurcated arteries, providing valuable insights for diagnosing and treating cardiovascular diseases.

This study aims to present a unique hybrid metaheuristic approach to solving the nonlinear analysis of hall currents and electric double layer (EDL) effects in multiphase wavy flow by merging the firefly algorithm (FA) and the water cycle algorithm (WCA).

Nonlinear Hall currents and EDL effects in multiphase wavy flow are originally described by partial differential equations, which are then translated into an ordinary differential equation model. The hybrid FA-WCA technique is used to take on the optimization challenge and find the best possible design weights for artificial neural networks. The fitness function is efficiently optimized by this hybrid approach, allowing the optimal design weights to be determined.

The proposed strategy is shown to be effective by taking into account multiple variables to arrive at a single answer. The numerical results obtained from the proposed method exhibit good agreement with the reference solution within finite intervals, showcasing the accuracy of the approach used in this study. Furthermore, a comparison is made between the presented results and the reference numerical solutions of the Hall Currents and electroosmotic effects in multiphase wavy flow problem.

This comparative analysis includes various performance indices, providing a statistical assessment of the precision, efficiency and reliability of the proposed approach. Moreover, to the best of the authors’ knowledge, this is a new work which has not been explored in existing literature and will add new directions to the field of fluid flows to predict most accurate results.

The purpose of this study is to investigate the effects of entropy generation of some embedded thermophysical properties on heat and mass transfer of pulsatile flow of non-Newtonian nanofluid flows between two porous parallel plates in the presence of Lorentz force are taken into account in this research.

The governing partial differential equations (PDEs) were nondimensionalized using suitable nondimensional quantities to transform the PDEs into a system of coupled nonlinear PDEs. The resulting equations are solved using the spectral relaxation method due to the effectiveness and accuracy of the method. The obtained velocity and temperature profiles are used to compute the entropy generation rate and Bejan number. The influence of various flow parameters on the velocity, temperature, entropy generation rate and Bejan number are discussed graphically.

The results indicate that the energy losses can be minimized in the system by choosing appropriate values for pertinent parameters; when thermal conductivity is increasing, this leads to the depreciation of entropy generation, and while this increment in thermal conductivity appreciates the Bejan number, the Eckert number on entropy generation and Bejan number, the graph shows that each time of increase in Eckert will lead to rising of entropy generation while this increase shows a reduction in Bejan number. To shed more light, these results were further demonstrated graphically. The current research was very well supported by prior literature works.

All results are presented graphically, and the results in this article are anticipated to be helpful in the area of engineering.

The purpose is to explore the free vibration behaviour of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The goal of this study is to gain a better knowledge of the dynamic response of nanoscale structures made of functionally graded materials and porous features. The Rayleigh-Ritz approach is used in this study to generate realistic mathematical models that take elastic foundation support into account. This research can contribute to the design and optimization of advanced nanomaterials with potential applications in engineering and technology by providing insights into the influence of material composition, porosity and foundation support on the vibrational properties of nanoplates.

A systematic methodology is proposed to evaluate the free vibration characteristics of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The study began by developing the mathematical model, adding material properties and establishing governing equations using the Rayleigh-Ritz approach. Numerical approaches to solve the problem are used, using finite element methods. The results are compared to current solutions or experimental data to validate the process. The results are also analysed, keeping the influence of factors on vibration characteristics in mind. The findings are summarized and avenues for future research are suggested, ensuring a robust investigation within the constraints.

The Rayleigh-Ritz technique is used to investigate the free vibration properties of elastic foundation-supported porous functionally graded nanoplates. The findings show that differences in material composition, porosity and foundation support have a significant impact on the vibrational behaviour of nanoplates. The Rayleigh-Ritz approach is good at modelling and predicting these properties. Furthermore, the study emphasizes the possibility of customizing nanoplate qualities to optimize certain vibrational responses, providing useful insights for engineering applications. These findings expand understanding of dynamic behaviours in nanoscale structures, making it easier to build innovative materials with specific features for a wide range of industrial applications.

The novel aspect of this research is the incorporation of elastic foundation support, porous structures and functionally graded materials into the setting of nanoplate free vibrations, utilizing the Rayleigh-Ritz technique. Few research have looked into this complex combo. By tackling complicated interactions, the research pushes boundaries, providing a unique insight into the dynamic behaviour of nanoscale objects. This novel approach allows for a better understanding of the interconnected effects of material composition, porosity and foundation support on free vibrations, paving the way for the development of tailored nanomaterials with specific vibrational properties for advanced engineering and technology applications.

The particle distribution in a fluid is mostly not homogeneous. The inhomogeneous dispersion of solid particles affects the velocity profile as well as the heat transfer of fluid. This study aims to highlight the effects of varying density of particles in a fluid. The fluid flows through a wavy curved passage under an applied magnetic field. Heat transfer is discussed with variable thermal conductivity.

The mathematical model of the problem consists of coupled differential equations, simplified using stream functions. The results of the time flow rate for fluid and solid granules have been derived numerically.

The fluid and dust particle velocity profiles are being presented graphically to analyze the effects of density of solid particles, magnetohydrodynamics, curvature and slip parameters. Heat transfer analysis is also performed for magnetic parameter, density of dust particles, variable thermal conductivity, slip parameter and curvature. As the number of particles in the fluid increases, heat conduction becomes slow through the fluid. Increase in temperature distribution is noticed as variable thermal conductivity parameter grows. The discussion of variable thermal conductivity is of great concern as many biological treatments and optimization of thermal energy storage system’s performance require precise measurement of a heat transfer fluid’s thermal conductivity.

This study of heat transfer with inhomogeneous distribution of the particles in a fluid has not yet been reported.

This paper aims to investigate the flow of two-layered non-Newtonian fluids with different viscosities in an axisymmetric elastic tube.

A mathematical model was considered for this study to describe the flow characteristics of two-layered non- Newtonian Jeffrey fluids in an elastic tube. Because Jeffrey fluid model is a better model for the description of physiological fluid motion. Further, this model is a significant generalization of Newtonian fluid model. Analytical expressions for flux, stream functions, velocities and interface velocity have been derived in terms of elastic parameters, inlet, outlet and external pressures. The effects of various pertinent parameters on the flow behavior have been studied.

The volumetric flow rate was calculated by different models of Mazumdar (1992) and Rubinow and Keller (1972); from this it was found that the flux of Jeffrey fluid is more in the case of Rubinow and Keller model than Mazumdar. A comparative study is made between single-fluid and two-fluid models of Jeffrey fluid flows and it was observed that more flux and higher velocities were observed in the case of two-fluid model rather than single-fluid model. Furthermore, when both the Jeffrey parameter tends to zero and ratios of viscosities and radii are unity, the results in this study agree with those of Rubinow and Keller (1972).

To describe the fluid flow in an elastic tube with two-layered systems, the models and solutions developed here are very important. These results will be highly suitable in analyzing the rheological characteristics of blood flow in a small blood vessel because of their elastic nature.

The aim of this research is to understand how government incentives (financial and non-financial) influence the relationship between green innovation and Sustainable Development Goals (SDGs) in SMEs.

To contribute to the literature, this research uses empirical evidence of 204 Pakistani small and medium-sized enterprises (SMEs) and tests the moderating role of government support between green innovation and SDGs.

The findings indicate that green innovation has a significant influence on SDGs, community development and environmental activities. The government support significantly strengthens the relationship between green innovation and environmental practices, while it does not moderate the path between green innovation and community development.

The research recommends SMEs focus on the adoption of green innovation and green technology to protect the environment and facilitate the community. Moreover, the research advises the government to assist SMEs financially and nonfinancially, so they will in turn help in the attainment of SDGs.

This research is the first attempt to assess the importance of green innovation in SDGs with a moderating role of government incentives in emerging SMEs. It provides several useful implications for policymaking.

The purpose of this study is to analyze the impact of inclined magnetohydrodynamics (MHD) and thermal radiation on the flow of a ternary micropolar nanofluid on a sheet that is expanding and contracting while applying mass transpiration and velocity slip conditions to the flow. The nanofluid, which is composed of Au, Ag and Cu nanoparticles dispersed in water as the base fluid, possesses critical properties for increasing the heat transfer rate and is frequently used in manufacturing and industrial establishments.

The set of governing nonlinear partial differential equations is transformed into a set of nonlinear ordinary differential equations. The outcome of this differential equation is solved and obtained the closed-form solution and energy equation in the form of hypergeometric functions.

The velocity, micro-rotation and temperature field are investigated versus a parametric variation. The physical domains of mass suction or injection and micropolar characteristics play an important role in specifying the presence, singleness and multiplanes of exact solutions. In addition, many nondimensional characteristics of the profiles of temperature, angular velocity and velocity profiles are graphically shown with substantial consequences. Furthermore, adding nanoparticles increases the heat transfer rate of the fluid used in manufacturing and industrial establishments. The current findings may be used for better oil recovery procedures, smart materials such as magnetorheological fluids, targeted medicine administration and increased heat transmission. Concerning environmental cleanup, nanomaterial fabrication and biomedical devices, demonstrate their potential influence in a variety of disciplines.

The originality of this paper is to analyze the impact of inclined MHD at an angle with the ternary nanofluid on a micropolar fluid over an expanding and contracting sheet with thermal radiation effect.